Granola mix P is made of dried fruit and grains in a ratio of by weight. Granola mix Q is made of dried fruit and grains in a ratio of by weight. Equal weights of mixes P and Q are thoroughly combined to produce a new mix R.
What fraction of the weight of mix R is dried fruit?
For mixture and ratio problems, first convert each ratio into a clear fraction of the total (fruit out of total parts). When equal amounts are combined, pick a convenient common total weight (often the least common multiple of the denominators) so all amounts stay as whole numbers. Compute the actual quantity of the target component from each source, add them, then divide by the new total weight. Finally, check that your result is reasonable—here it should lie between the original two fruit fractions and be well below 1.
Hints
Turn each ratio into a fraction
For a ratio such as (fruit to grains), how many total parts are there, and what fraction of the total parts are fruit? Do the same for mix Q's ratio, .
Use equal weights with easy numbers
Since equal weights of the two mixes are combined, you can imagine using any convenient equal weights, such as units each. Pick a number that is a common multiple of the denominators of the fruit fractions you found.
Add fruit amounts, then compare to total weight
Find how many units of fruit come from each mix at your chosen weight, add them, and then divide this total fruit weight by the total weight of both mixes together to get the fraction that is fruit.
Desmos Guide
Confirm fruit fractions
In Desmos, enter 3/10 and 5/16 to verify the fractions of fruit for mixes P and Q (you can compare their decimal values if that helps your intuition).
Compute the fraction of fruit in the combined mix
In Desmos, type the expression (3/10*80 + 5/16*80)/160. The calculator’s output is the fraction of the total weight of mix R that is dried fruit; compare this value to the answer choices.
Step-by-step Explanation
Convert each ratio into a fruit fraction
For each mix, the ratio gives the amount of fruit relative to grains.
- Mix P has ratio . That means there are total parts, and of them are fruit, so the fraction of fruit is .
- Mix Q has ratio . That means there are total parts, and of them are fruit, so the fraction of fruit is .
Assume convenient equal weights for mixes P and Q
Because equal weights of mixes P and Q are combined, you can choose any convenient same weight for each mix. Pick units for each mix because is a common multiple of and , which keeps everything in whole numbers.
Find the amount of fruit from each mix and the totals
Compute how much fruit comes from each -unit batch.
So, total fruit in mix R is units.
Total weight of mix R is units.
Compute the fraction of mix R that is fruit
Now form the fraction
So the correct answer is , which corresponds to choice D.